Behavioral Experiments on a Network Formation Game

Michael Kearns, University of PennsylvaniaStephen Judd, University of PennsylvaniaYevgeniy Vorobeychik, Sandia National Laboratories

Abstract: We report on an extensive series of behavioral experiments in which 36 human subjectscollectively build a communication network over which they must solve a competitive coordination task formonetary compensation. There is a cost for creating network links, thus creating a tension between linkexpenditures and collective and individual incentives. Our most striking finding is the poor performanceof the subjects, especially compared to our long series of prior experiments. We demonstrate that thesubjects built difficult networks for the coordination task, and compare the structural properties of thebuilt networks to standard generative models of social networks. We also provide extensive analysis ofthe individual and collective behavior of the subjects, including free riding and factors influencing edgepurchasing decisions.

Categories and Subject Descriptors: Computer Applications [Social and Behavioral Sciences]: Eco-nomics, Psychology, Sociology

General Terms: Economics, Experimentation, Human Factors, Theory

Additional Key Words and Phrases: Social Networks, Game Theory, Network Formation

1. INTRODUCTIONIn recent years, research from a variety of disciplines has established the universalityof certain approximate structural properties of large-scale social, technological, orga-nizational and economic networks. These properties include networks having smalldiameter, high clustering of connectivity, and heavy-tailed degree distributions. Theapparent ubiquity of these properties, despite the diversity of the domains of the net-works in which they appear, has led researchers to seek explanations in the form ofmodels of network formation that can reliably generate the observed structures.

The most studied class of such models are stochastic network formation models, inwhich networks form through a decentralized process that generates local connec-tivity using randomization; examples include the classic Erdos-Renyi random graphmodel [Bollabas 2001], and the more recent small worlds [Watts and Strogatz 1998;Kleinberg 2000] and preferential attachment [Barabasi and Albert 1999] models.These models have been successful in providing simple and relatively general mecha-nisms generating common structural properties of large networks.

An important criticism of the stochastic formation models is that in real networks,connectivity rarely forms entirely randomly — rather, there is often some significantcomponent of purposefulness when a new node or link is formed. Professionals joinand form connections on a service like LinkedIn to enjoy the career benefits of beingpart of that network; new web sites are created to generate traffic, including by being

This work was conducted while Y.Vorobeychik was at the University of Pennsylvania.Permission to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrightsfor components of this work owned by others than ACM must be honored. Abstracting with credit is per-mitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any componentof this work in other works requires prior specific permission and/or a fee. Permissions may be requestedfrom Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212)869-0481, or permissions@acm.org.EC’12, June 4–8, 2012, Valencia, Spain. Copyright 2012 ACM 978-1-4503-1415-2/12/06...$10.00.

part of the network of links and the attendant search indexing benefits. While theremay be elements of arbitrariness or stochasticity, networks generally arise from theself-interests of their constituents, and serve some collective purpose(s).

An alternative class of economic or game-theoretic models directly addresses the is-sue of self-interest and purpose in network growth. In network formation games [Tar-dos and Wexler 2007; Jackson 2005; 2010; Fabrikant et al. 2003; Borgs et al. 2011;Albers et al. 2006; Brautbar and Kearns 2011; Even-Dar et al. 2007; Even-Dar andKearns 2006], individual players typically have utility functions with two competingcomponents: there is a cost to join the network, usually in the form of purchasinglinks to other players (where the cost may be viewed as monetary, as in the connec-tivity and physical costs of adding a router to the Internet, or more cognitive, as inthe time needed to create and maintain friendships on Facebook). But after joiningthe network, a player enjoys participation benefits (perhaps abstracted by some mea-sure of their centrality in the network), and their overall payoff is the network benefitminus their cost of joining. It is common to equate the outcome of such games withtheir Nash equilibria, just as the stochastic models are analyzed for their statisticallytypical properties. While there has been growing interest in the theory of network for-mation games for several years now, to our knowledge there is not an accompanyingbehavioral literature.

In this paper, we describe among the first and largest human-subject experiments ina pure network formation game. These are the most recent in a long series of behav-ioral experiments on strategic and financial interactions in social networks [Kearnset al. 2006; Kearns et al. 2009; Judd et al. 2010; Kearns and Judd 2008; Chakrabortyet al. 2010; Kearns et al. 2011], but represent a major departure from our prior ex-periments, where the networks examined were always exogenously imposed on thesubjects. Here we endogenized the formation of the network structure itself as partof the experiment. While the theoretical literature on network formation games hasfocused on one-shot, simultaneous move games of full information (and even there,characterizations of equilibria are difficult and elusive), our experiments investigate aformation game of continuous, asynchronous moves and partial information.

In the experiments, subjects were given financial incentives to solve a collective butcompetitive coordination problem of biased voting, in which they must unanimouslyagree on one of two alternative choices, or receive no payoff at all. The competitiveaspect arises from the fact that different players have different financial preferencesfor which of the two choices is agreed upon. We have previously studied this prob-lem on fixed network structures [Kearns et al. 2009]; in the current experiments thesubjects themselves had to build the network during the experiment, via individualplayers purchasing links whose cost is subtracted from their eventual task payoff. Thenature of the biased voting task and the financial self-interests of the players sets upa clean strategic tension: in order to solve the biased voting problem, the players mustcollectively purchase enough links to establish some minimal global connectivity; butany individual player would prefer others to incur the costs of building this sharedinfrastructure.

A striking finding is that the players performed very poorly compared to our long se-ries of prior experiments in which network structures were imposed exogenously. De-spite clearly understanding the biased voting task, and being permitted to collectivelybuild a network structure facilitating its solution, subjects instead appear to have builtvery difficult networks for the task. This finding is in contrast to intuition, case stud-ies and theories suggesting that humans will often organically build communicationnetworks optimized for the tasks they are charged with, even if it means overridingmore hierarchical and institutional structures [Burns and Stalker 1994; Nishiguchiand Beaudet 2000]. We also report on a number of other aspects of subject behavior

and performance, including structural properties of the built networks, comparisonsto standard network formation models, and free-riding in edge purchasing.

2. EXPERIMENTAL DESIGN AND METHODOLOGYThe experiments reported here were held in three sessions with different pools of 36subjects each. As in our previous experiments, subjects sat at networked workstationsseparated by physical partitions, and the only communication permitted was throughthe system. In each of many short experiments, subjects were given financial incen-tives to solve a global coordination problem via only local interactions in a network.In prior experiments, the network structure was a design variable that we chose andimposed exogenously in each experiment; here, the network structure was created bythe subjects themselves, as described below.

We first describe the overarching collective task the subjects were charged with solv-ing. In each experiment, subjects sat at individual workstations, and each controlledthe state of a single vertex in a 36-vertex network whose connectivity structure evolvedthroughout the experiment. The state of a subject’s vertex was simply one of two colors(red or blue), and could be asynchronously updated as often as desired during the one-minute experiment. Subjects were able to view the current color choices of only theirimmediate neighbors in the current network at all times. No communication betweensubjects outside the experimental platform was permitted.

In each experiment, each subject was given a financial incentive that varied acrossthe population, and specified both individual preferences and the demand for collec-tive unity. For instance, one player might be paid $2 for blue consensus and $1 forred consensus, while another might be paid $1 for blue consensus and $2 for red con-sensus, thus creating distinct and competing preferences across individuals. However,payments for an experiment were made only if (red or blue) global unanimity of colorwas reached; thus subjects had to balance their preference for their higher payoff colorwith their desire for any payoff at all.

At the beginning of each 1-minute experiment, the network over the players was typ-ically empty: there were no edges, and thus every player controlled an isolated vertexand could see only their own color choice. Clearly reaching unanimity of color choicein the biased voting task is highly unlikely in such circumstances. Thus at any timethroughout an experiment, subjects were free to purchase edges to other players ata fixed cost. Edge purchases were unilateral — they did not require approval by theplayer on the receiving end — but their benefits were bilateral, meaning that after thepurchase both players could see each other’s current color choices. The system GUI (seeFigure 1) would dynamically evolve as new edges were purchasing, always showing asubject the color choices of their neighbors in the current network.

An important design decision is what information the players are provided to helpthem decide which vertices to purchase edges to. One possibility is no information:players could simply indicate their desire for a new connection, and the system couldsimply give them a new edge to a random player to whom they were not alreadyconnected. However, this would predetermine the network topologies built to be ofa random, unstructured, Erdos-Renyi variety, and not particularly useful for the bi-ased voting task. We thus decided to give subjects two pieces of information abouttheir current non-neighbors: their current degree (number of connections), and theircurrent shortest-path distance in the network from the subject. This allowed players toselectively purchase edges that seem relevant to the task. For instance, players couldchoose to buy edges to players with high degree who were distant from them in thecurrent network (perhaps in the hopes that such players aggregate information on theother side of the network), or to players with zero or low degree (perhaps in the hopesof having strong influence over the color choice of such players). We emphasize that

Fig. 1. Sample screenshot of player GUI in the network formation game. Around the vertexlabeled “YOU”, the central panel displays the player’s current network neighborhood, indicatingthe current color of their neighbors as well as any edges between neighbors. In the action panelat the bottom, the player can change their own current color by clicking on the buttons labeled“red” and “blue”. To the lower left of the central panel is the grid where players can selectother vertices to purchase edges to. Each non-neighbor is represented by a circle whose gridposition indicates their current degree and shortest-path distance from the player. (Verticesnot currently in the same connected component as the player are shown as being at infinitedistance.) If more than one vertex has the same degree and distance, the circle contains a “*”symbol. The player purchases edges by clicking on the desired circle, at which point their newneighbor will be incorporated into the neighborhood display. At the top, the time elapsed in theexperiment is shown, along with the payoffs of the player, which are dynamically reduced asedges are purchased. The fixed cost per edge is also displayed.

this choice of informational design was not made with realism and generality in mind— obviously, in real social networks one does not have knowledge of the degree anddistance of non-neighboring vertices — but rather potential relevance to the collectivetask, which seemed to us a more important experimental criterion.

We also note that this informational design also permitted the subjects, in principle,to collectively generate networks similar to those of well-studied stochastic modelssuch as Erdos-Renyi (by simply ignoring the degree and distance values, and alwayschoosing a random non-neighbor to connect to), or networks similar to those generatedby preferential attachment (by ignoring distance information, and favoring purchasingedges to higher-degree vertices). We know from previous fixed-network experimentsthat such models generate networks generally favorable for human performance across

a wide variety of tasks, including biased voting [Kearns et al. 2009]. Thus at least somebehaviorally “easy” networks are collectively reachable within the given system design.

Each player’s GUI had an edge-purchase panel in which each of their current non-neighbors was represented by a circle on a 2-dimensional grid, indicating that non-neighbor’s current degree and distance from the player; see Figure 1. By simply click-ing on the corresponding circle, a player would purchase an edge, and the new neigh-bor and their color would be dynamically incorporated into their network neighbor-hood display, and remain for the duration of the experiment. (Edge purchases werepersistent and irrevocable.) If more than one non-neighbor had the same degree anddistance, the grid circle would indicate so. As an experiment progressed, degrees in-creased and distances decreased in the growing network.

If the players failed to reach unanimity in the allotted time, all edge purchases wereforgiven, and no payoffs were made; but if unanimity was reached at any point, theexperiment was terminated, and a player’s edge purchases were subtracted from theirearnings on the biased voting problem to arrive at their net payoff. The system alsoenforced the condition that players must have strictly positive payoffs on successfulexperiments: thus, each player could only spend an amount on edges that was slightlyand strictly less than their lower-payoff color in the biased voting problem. This pre-vents players from becoming “infinitely stubborn” in favor of their higher-payoff colorif their lower payoff has been reduced to zero by edge purchases.

In a subset of experiments, conditions were as described above, but instead of start-ing with the empty network (which we shall refer to as unseeded experiments in thesequel), the experiment began with a “seed” network of edges that were provided freeof charge to the players [Kleinberg 2000; Even-Dar and Kearns 2006]; players couldthen optionally purchase additional edges as above. Thus each experiment was char-acterized by the distribution of biased voting incentives of the players, the presenceor absence of the seed network and its structure, and the fixed price of edges (whichwe varied from experiment to experiment). We shall comment on each of these designvariables in the appropriate places as we describe our findings.

From a theoretical perspective, we thus presented our subjects with a task-orientednetwork formation game of partial information (unknown incentives or types of theother players, unknown and evolving global network structure) and asychronous, re-peated moves with finite termination time. We note that formal analysis of even vastlysimplified versions of this game appears to be quite challenging, but might be an in-teresting avenue for future work.

3. BACKGROUND ON PRIOR FIXED-NETWORK EXPERIMENTSAs mentioned above, we have conducted experiments similar to those described heresince 2005, but always designing and exogenously imposing the network structuresmediating interaction. The tasks we have given to subjects are diverse, and includegraph coloring [Kearns et al. 2006], consensus [Judd et al. 2010], networked trad-ing [Kearns and Judd 2008], networked bargaining [Chakraborty et al. 2010], inde-pendent set [Kearns et al. 2011], and biased voting [Kearns et al. 2009]. While directcomparisons across tasks can be difficult, there is one general and easily measuredmetric of collective performance, which is the efficiency: in any given experiment, wecan compute the configuration of play that would have maximized the total paymentsto the subjects, and then compute the fraction of that maximum payoff the popula-tion actually realized. We can then average this quantity across all experiments everconducted, regardless of task, network structure, and other design variables. The re-sulting value is 0.88 — in other words, over the lifetime of the project, subjects haveextracted almost 90% of the value that was available to them in principle. We conclude

that humans are quite good at solving a variety of challenging tasks from only localinteractions in an underlying network.

In the previous biased voting experiments on exogenous networks, 55 of 81 exper-iments resulted in unanimity and therefore some payoff to all subjects, again givingfairly strong collective performance. On the subset of experiments in which the net-work and incentives had what we called a minority power structure, performance waseven stronger, with 24 of 27 experiments reaching consensus. We shall contrast thesefindings with those for biased voting with network formation.

4. RESULTS4.1. Overall PerformanceOur experiments were structured in three separate sessions with different subjectpools; while the conditions in the first two sessions were similar and designed in ad-vance, the third session was designed in response to the findings of the first two, andshall be discussed separately below.

Session 1 consisted of 99 short experiments, with 63 of these being unseeded; Ses-sion 2 consisted of 72 experiments, 27 of which were unseeded. Across these 171 exper-iments, various other conditions varied as well, including the cost per edge, the fractionof players with higher payoffs for red, and the relative strengths of the incentives forplayers of different types. We shall discuss the effects of these design variables later,and for now focus on the collective performance across all these conditions in Sessions1 and 2.

Compared to our long series of prior fixed-network experiments, that performancewas surprisingly poor: Session 1 produced only 47% successful (unanimous) outcomes,and Session 2 only 39%, for an overall success rate of 44%. This is in sharp contrastto the aforementioned efficiency across all tasks of 88% — approximately double thatof the current experiments — and the 68% success rate of the fixed-network biasedvoting experiments, more than 20% higher than in the network formation game. Itappears that allowing the subjects to control the creation of the network significantlyworsened collective performance 1.

There are at least two plausible explanations for this degradation in performancethat do not simply entail that subjects built “bad” networks for the task. The firstexplanation is one of cognitive overload: perhaps subjects built “good” networks, butsimply ran out of time to solve the biased voting problem on those good networks.The second explanation is one of stubbornness due to modified incentives: perhapsthe subjects built good networks, but due to the edge purchases, some players hadreduced the net payoff of their less preferred color to such a small amount that theymight be very resistant to acquiesce to the majority color, resulting in stalemates.This hypothesis is made more plausible by the significant amount of “free riding” thatoccurred with respect to edge purchases, discussed later.

To investigate the overload and stubbornness hypotheses, we designed and con-ducted a third session of experiments with fresh subjects. In Session 3, each gamewas seeded with a network that was the final network constructed by the subjects inan unseeded Session 1 experiment. In these Session 3 experiment, the subjects onlyplayed the biased voting game — no edge purchases were allowed by the system. Thesubjects were thus back in the setting of our earlier, exogenous fixed-network experi-ments, but this time using networks built by previous human subjects. We deliberatelychose a subset of the final networks from Session 1 on which the performance therewas particularly poor — namely, 18 final networks on which the success rate was only

1In the games that failed to converge, the average size of the minority was actually smaller than it was forthe failed biased voting games, but not to the level of statistical significance.

Session  1:  

Ini+al  (empty)  network  and  rela+ve  incen+ves  

Final  (built)  network  and  modified  incen+ves  

Session  3,  Fixed  Networks:  

Network  built  in  Session  1,  ini+al    incen+ves  restored  

Network  built  in  Session  1,  modified  incen+ves  preserved  

Fig. 2. Design of Session 3 experiments. At the start of an unseeded Session 1 experiment (left),the network is empty and some players (the upper 3) have higher payoffs for red than blue,schematically represented by the ratio of the red to blue areas in each vertex. At the completionof the experiment (middle), a network has been built by the players, and some players may havemore extreme relative preferences due to the reduction of their payoffs by edge purchases. InSession 3 (right), we took the final networks built by Session 1 experiments, and exogenouslyimposed them as the networks of a pure biased voting game without any edge purchases. Wedid so using both the original, restored incentives of Session 1, and the post-edges modifiedincentives.

17%. We ran each Session 3 network under two different incentive conditions: one inwhich the incentives the players were the same as at the beginning of the correspond-ing Session 1 experiment, and one in which they were the same as at the end (afteredge expenditures). See Figure 2 for a description of Session 3 design.

Together these conditions allow us to investigate the validity of the explanationsabove: if subjects were simply running out of time in Session 1 (the overload hypothe-sis), they should fare much better in Session 3, since now the network formation taskis removed, and they can focus only on the biased voting task; and if the difficulty inSession 1 was due to stubbornness after edge purchases, the Session 3 experiments inwhich the Session 1 networks are used but the incentives are restored to their startingvalues should be more successful.

The stubbornness and overload hypotheses are strongly refuted by the Session 3 re-sults. Success rates were slightly higher than in Session 1, but not significantly so.The strong signal was that Session 3 success rates in games using the original payoffs,and in games using residual payoffs, were both significantly lower than in our earlierfixed-network biased voting games [Kearns et al. 2009], with P < 0.01 in both cases.The success rate (57/81) during all games in the fixed-network biased voting sessionwas significantly higher than that of all games (100/243) in the three network forma-tion sessions (P < 0.0001). More pointedly, the games in each of the three networkformation sessions are individually lower than the fixed-network biased voting games(all with P < 0.01).

We are thus led to the conclusion that our subjects built networks on which it wassimply difficult to accomplish the very task they were being paid to accomplish. Theyappear to have had enough time and incentive, but built inherently poor networks. Asper the earlier discussion, this is the first task of the many we have investigated inwhich human performance was so low.

4.2. Effects of Seed NetworksRecall that a subset of the Session 1 and 2 experiments explored network formationin which the subjects were provided an initial seed network as free infrastructure.Our goal was to examine whether having this seed network, which might facilitatecommunication and coordination at no cost, would allow the players to build betternetworks and yield stronger performance.

We examined three types of seed network structure: a 2-dimensional grid or torusnetwork, which provides global connectivity with relatively few edges; a network of6 cliques of size 6, which groups the players into small highly-connected communi-ties; and preferential attachment networks that were also the focus of the minoritypower experiments we discuss shortly. Again, in each of these experiments the playerswere free to purchase additional edges that would be dynamically added to the seednetwork.

Again somewhat surprisingly, none of these seed network structures seemed to im-prove collective performance much, with the completion rates on seed torus experi-ments being 33% (6/18), and on seed clique experiments being 33% (9/27) — neitherof which is significantly different from the unseeded networks of Session 1, but bothof which are significantly lower for the previous fixed-network biased voting networks(P < 0.01). Whatever network the subjects were given to start, they seem to haveturned it into a poor network for the task.

A major finding of our original biased voting experiments focused on experiments inwhich networks were generated according to preferential attachment, which results ina heavy-tailed degree distribution, and where we gave a (sometimes very small) minor-ity of the players a higher payoff for red, with the majority preferring blue. The twistwas that the red minority consisted of the highest-degree vertices in the network; wewere thus investigating whether a small but well-connected minority could system-atically impose its preference against the majority’s. The answer was resoundinglypositive: 24 of 27 (89%) such fixed-network experiments ended in consensus, every oneof them on the minority preference [Kearns et al. 2009].

In the current experiments, we were interested in how this finding might be changedif subjects could add edges to the minority power networks. We thus ran a number ofexperiments in which both the seed network structure and arrangement of incentiveswere identical to those in the earlier minority power experiments. The success ratewas 61% (22/36) — higher than for torus or clique seeds, but still much lower than theoriginal exogenous network minority power rate of 89%. Once again, permitting thesubjects to modify the network has harmed collective performance. However, now 35%of the successful experiments ended with the majority preference — compared withnone in the exogenous network case, a dramatic change. One interpretation is thatpermitting the purchase of edges allows the majority players to better realize theyare in the majority — which may have been difficult in the exogenous network case,especially for the preponderance of low-degree vertices — and causes them acquiesceto the minority less readily. This could account both for the lower overall success rate,and the increased rate of majority victories.

4.3. Effects of Edge Costs and IncentivesRecall that across the many experiments, we varied the cost per edge purchase, andthe absolute and relative incentives across the population. Edge costs were low ($0.01),medium ($0.10), or high ($0.25), with the proviso that the edge purchases of a playermust always be strictly less than $1 (which was always the payoff of the less preferredcolor).

Not surprisingly, the cost per edge had a strong effect on the resulting network den-sity (as we shall see in the following section), but also on collective performance. Theoverall success rates for unseeded Session 1 experiments were 67% for low-cost ex-periments, 38% for medium-cost experiments, and 14% for high-cost experiments; thedifferences between these quantities are all significant at P < 0.05. We note that al-though there was a clear relationship between edge costs and performance, with highercosts resulting in worse performance, in no case did the subjects collectively approachthe maximum allowed edge expenditures; the fraction of possible edge purchases inunseeded Session 1 experiments was 64% for low edge cost, 42% for medium, and 59%for high. Thus subjects could have built considerably denser networks in all cases,but chose not to. We also note that the seeded experiments provided the subjects withconsiderably denser networks for less edge expenditures, yet failed to significantly im-prove performance.

The incentives given to players also had a pronounced effect on subject performance.Recall that in each experiment a subject always desired unanimity (no payment wasdistributed unless unanimity was reached), but had a preference for one color overanother. For example, a subject might receive 4 if all chose blue, and only 1 if theconsensus was to red. We maintained the smaller incentive at 1 for all experiments,and varied the payoff of the preferred color, from 4 (strong preferences), 2 (weak pref-erences), and 1 (indifference between the color choices). The overall success rates forSession 1 experiments were 58% (7/12) when the subjects were indifferent between thetwo colors, 53% (19/30) when they had a weak preference for one of the colors, and only17% (4/24) when they had a strong color preference. While there was no statisticallydetectable difference between the impact of weak and no preference, strong preferencehad a clear detrimental effect on solution rate (P < 0.01, comparing against weakpreference).

4.4. Network Structure and Centrality SkewnessBoth as a general matter, but especially in light of the overall poor performance ofthe subjects, the structure of the networks built during the experiments is a topic ofcentral interest. How do the built networks compare to more naturally evolved socialnetworks, and what properties of the built networks might account for the difficultythey posed for the biased voting problem? Here we initially restrict our analysis to themajority of experiments where there was no seed network, so that all structure wasbuilt by the subjects themselves.

We begin by establishing that the built networks actually do share a number of struc-tural “universals” that appear frequently in real-world networks. The first remarkworth making is that in every unseeded experiment, the subjects built a connectednetwork — there were never two or more disconnected components. Thus regardlessof the edge costs and other parameters, the subjects always bought enough edges toestablish global communication.

The diameters (pairwise average shortest path distances) of the built networks weregenerally quite small compared to the population size; while the diameters show astrong dependence on the network density and therefore the edge costs, they averaged1.32 for all low-cost experiments (standard deviation 0.17), 1.87 (standard deviation

Fig. 3. Clustering coefficients vs. edge density for networks built in all unseeded experiments.For each built network, the x value indicates the fraction of possible edges present, and the yvalue indicates the clustering coefficient of the network. Random networks of the same densitieswould have clustering coefficients equal to their density, as suggested by the diagonal line. Wesee that clustering in the built networks is uniformly higher except at very low densities, wherepresumably the subjects may be been primarily concerned with establishing global connectivity.There are strong effects of edge costs (coded by color); higher edge costs consistently lead tolower densities and clustering. The mixed experiments had variable edge costs for the players,but always either medium or high.

0.19) in medium-cost experiments, and 2.38 (standard deviation 0.16) in high-cost ex-periments. Also as is for typical social networks, the clustering coefficients of the builtnetworks were generally much higher than for random networks of the same density;see Figure 3. Furthermore, examination of the degree distributions of the built net-works reveals the presence of “connector” vertices whose degrees are several timeslarger than the mean, another commonly cited property of natural social networks.

Given that the structural properties cited so far were generally present in our earlier,fixed-network experiments — in which the subjects performed much better — whatcan account for the difficulty of the built networks? While it is impossible to be certain,due to the complexities of both the networks and subject behavior, a strong candidateis the overreliance on very few vertices or subjects for connectivity and communica-tion. In Figure 4 we demonstrate this reliance visually by showing, for three of thebuilt networks, the effects on structure and connectivity of deleting a few vertices withthe highest degrees. In each case, the network quickly becomes highly fragmented, aproperty generally true of the built networks.

We can make this analysis more systematic and rigorous by considering the quantityknown as betweenness centrality (centrality in the sequel). Centrality is designed tomeasure, for each vertex u, the extent to which the rest of the network relies on u forits global connectivity and communication. More formally, we define

CB(u) =∑

v,w∈V :v 6=u,w 6=u,v 6=w

nuv,w

nv,w

Fig. 4. Visualization of built networks in three unseeded Session 1 experiments with high edgecosts. The first column shows a visualization of each of the built networks. Subsequent columnsshow the networks after repeated deletions of the highest-degree vertex remaining. In eachcase, the first deletion already shatters the network into multiple connected components, andsubsequent deletions yield a large number of isolated vertices.

where V is the set of all vertices, nv,w is the number of shortest paths between v andw, and nu

v,w is the number of shortest paths between v and w that pass through vertexu. Thus CB(u) is a global measure of how often vertex u appears on shortest pathsbetween all pairs of other vertices; it is a common metric of influence on communicationand connectivity in social networks.

Echoing the analyses above, it turns out that the subject-built networks systemat-ically differ from naturally occurring networks, and the ones we imposed on subjectsin our earlier experiments, in their distribution of centrality. In particular, as withdegrees, the built networks display a considerably more skewed distribution of CB(u):compared to natural network models at the same edge density, there are more ver-tices with very high and very low CB , and fewer with intermediate values of CB . SeeFigure 5.

There are a number of obvious reasons why overreliance on a few high-centralityvertices might make the biased voting task difficult. If a large fraction of the popu-lation implicitly relies on high centrality vertices to be effective aggregators of globalinformation (such as the current majority color), noisy or selfish behavior by these in-dividuals can impede collective performance. In the successful Session 1 experiments,the correlation between centrality and whether a subject received their higher pay-off was both positive (0.18) and highly significant (P < 0.001), suggesting that high-centrality players may have implicitly used their position to influence outcome rather

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Comparing built graphs with other graph types

solid: average of 21 graphs built by human subjects

dashed: average of 210 PA graphs with the same densities

dotted: average of 210 E-R graphs with the same densities

blue: High-cost, green: Medium-cost, red: Low-cost

Fig. 5. Comparison of the distribution of centrality CB between subject-built networks and stan-dard generative models. For a given network, we sort vertices in order of increasing CB values,and then compute the sum of all CB values through a given rank in the ordering. We then av-erage such curves over many built networks or many sampled networks from the generativemodels. We group the averaged curves by edge costs (color coded in the figure) for the built net-works, and compare them to Erdos-Renyi and preferential attachment networks of the sameaverage density. At each edge cost, we see that the built networks (solid lines) have much moreskewed distributions of CB than Erdos-Renyi (dotted lines) and preferential attachment (dashedlines): while the sum of all CB values (rightmost point) is comparable for all three classes of net-works at each density, much more of the cumulative centrality is accounted for by the final few,most central, vertices in the built networks, whose curves are considerably more convex thanfor the models.

than coordinate behavior — potentially contributing to the great majority of failedgames.

We note that one might be tempted to think that the starting seeds would dissipatethe propensity for skewness in the CB values. For instance, in the torus networks,there already are 4 edges uniformly assigned to each node, so one could imagine newedges would have less of a biasing effect than when starting from an empty network.However, this intuition is misleading. When we examined the CB values in networksthat were seeded, we found final values that were both higher and lower than thestarting values. The variance in people’s buying behavior injected a variance into theCB values, and they were spread out in both directions.

4.5. Purchasing Behavior and Free RidingThus far we have offered evidence that subjects built poor networks for the given task.It is natural to ask what particulars of subject behavior accounted for this. In thissection, we examine the distribution of edge purchases in the population, which shedssome light on this question. Here the most striking aspect is the preponderance of free-riding: at all edge costs and in each experiment, there is a significant fraction (roughly20% or more) of players who purchase no edges, and another large group who purchasevery little compared to the average. Thus the vast majority of the cost of building thenetworks was undertaken by only a small fraction of the population. This variancein behavior is what we believe generated the aforementioned variance in CB values.

For low and medium edge costs, fully 50% of the population contributed less than10% of the total edge expenditures; at high edge costs, where no player could affordto purchase more than 3 edges and thus variability of expenditure should be reduced,the free riding 50% still purchase less than 20% of the edges. Furthermore, free ridingwas an economically beneficial policy: at the level of individual human subjects, thecorrelation between the number of edges purchased in all Session 1 experiments andtheir total payoff is -0.72 (P < 0.001). The primary builders of the networks were thusapparently not financially favored by their resulting positions in it. Networks tendedto be built rapidly, with the vast majority of edge purchases coming in the first half ofthe allowed time.

Fig. 6. Frequencies of edge-purchasing decisions with respect to non-neighbor vertex distances(left panel) and degrees (right panel). For each edge purchase, the left axis represents how manydistinct choices the purchaser had, and the right axis represents which of these ranked optionsthey selected. The vertical axis then shows the relative frequency they made each ranked choice.Thus the diagonals indicate cases where they purchased an edge to the most distant, or highest-degree, non-neighbor, respectively. We thus see that while there is a tendency to purchase edgesto the most distant and highest degree vertices, there is also considerable mass at low andintermediate distances and degrees. Color is for visualization clarity only.

So far our emphasis has been on the structural properties of the built networks andthe distribution of expenditures; we next examine the criteria subjects seemed to use inedge-purchasing decisions, within the constraints of the degree and distance informa-tion they were provided about current non-neighbors. Normalization is an issue heresince (for instance) what constitutes a relatively “high degree” vertex is different nearthe start of an experiment than towards the end. Instead we can simply ask, at eachmoment an edge purchase was made, how many choices the purchaser had in eachdimension, and which one they made: that is, how many different degree values, andhow many different distance values, were populated by at least one non-neighbor onthe edge-purchasing GUI grid. The results are summarized in Figure 6, and they showthat while subjects most often chose to buy edges to vertices with the highest avail-able distance, or the largest available degree, they also frequently chose deliberatelylow values in both dimensions as well, and there is significant mass on intermediatevalues as well. Thus purchasing behavior is not easily consistent with standard gener-ative models such as Erdos-Renyi (which would induce uniform distributions in bothdimensions) or preferential attachment (which would not generate the observed ten-dency to connect to low-degree vertices). Despite the simplicity of the informationalinterface, subject purchasing behavior does not easily fall into simple models.

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Fig. 7. Edge purchasing rate (edges purchased per subject per second) as a function of the friend-liness quotient (see text) of the purchaser at the moment of purchase. The red curve shows theaggregate across all Session 1 and 2 purchases, and the peak near 0.5 is 60 or 70% higher thanat the edges. If we condition on the purchaser also having a current degree in some range (bluecurves annotated by degree range), we see that this tendency to purchase more when there is lo-cal indecision and conflict becomes greatly pronounced at higher degrees. The two higher degreecurves are 5 to 10 times higher in the middle than at the sides.

While the preceding analysis examines how subjects used degree and distance inedge purchasing decisions, it is also of interest to investigate how such decisions wereinfluenced by the local state of play — in particular, whether most neighbors wereplaying the subject’s preferred (higher payoff) color or not. Figure 7 shows the rate atwhich subjects purchased edges in Session 1 and 2 experiments as a function of the“friendliness quotient” of their neighborhood at the moment of purchase. The friend-liness quotient is the fraction of current neighbors who are playing the purchaser’shigher-payoff color. We see that there is a marked increase in the proclivity of a playerto buy an edge if she finds herself with an approximately equal number of friends andenemies (friendliness quotient 0.5). In situations where her neighbors are largely col-ored the same (whether of the higher or lower payoff color), she refrains from buying.This tendency becomes even more pronounced if we condition on just those purchasesmade by players whose current degree is higher. It may be that this tendency to buymore edges when there is a large amount of local disagreement (high degree, friendli-ness quotient close to 0.5) only worsened the indecision for everyone, thus leading topoor convergence performance.

5. CONCLUDING REMARKSThe results presented here have shown that human subjects, given the opportunityand incentives to collectively build a network in service of a competitive coordinationtask, did so poorly, creating networks inherently difficult for the task. This occurreddespite an edge purchasing interface that permitted, in principle, the creation of net-works known from earlier experiments to be much easier, such as random or prefer-ential attachment networks. Our findings are in contrast to some case studies and

theories about the abilities of human populations to effectively self-organize in serviceof collective goals. This contrast clearly deserves further scrutiny and controlled study.

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